Jones drove in the go-ahead run in the seventh inning and Soriano capped three scoreless innings by Atlanta's bullpen as the Braves beat the Mets 5-3 on Thursday night.
"He's been automatic, every time out," Jones, a six-time All-Star, said of Soriano. "He should have been an All-Star. I can't believe he didn't make it."
Jones' two-out fielding error in the ninth on Brian Schneider's grounder to third kept Soriano from his seventh 1-2-3 inning in his last nine appearances. He lowered his ERA to 1.44 and has 13 saves in 14 chances.
Soriano struck out pinch-hitter Jeremy Reed to end the game with a 96 mph fastball.
Catcher Brian McCann was the Braves' only All-Star in Tuesday night's game.
The Braves broke a 3-3 tie against the New York bullpen, which gave up two runs in two innings.
Pedro Feliciano (2-3) walked pinch-hitter Brooks Conrad, who advanced to third on Martin Prado's hit-and-run single to left off Bobby Parnell. Jones, a .330 career hitter against the Mets, drove in Conrad with a single to right off Parnell to give the Braves a 4-3 lead.
"You expect Chipper to deliver every time," Braves manager Bobby Cox said.
The win left the Braves (44-45) one game under .500 and 1 1/2 games ahead of the fourth-place Mets (42-46) in the NL East.
"This is a big series," Jones said. "It's a chance for us to play a good team. They're in our rearview mirror now. Hopefully we keep them there. You certainly want to get the second half kicked off on the right foot, and tonight was a good one."
Pinch-hitter Greg Norton added a run-scoring single off Pat Misch, New York's third reliever, in the eighth.
Mets manager Jerry Manuel acknowledged the team's first series after the All-Star break "is huge for us."
"We wanted to come out of the gate and play well," Manuel said.
"We didn't beat ourselves. We got a good effort from [Oliver] Perez. Hopefully, this will be a springboard for the rest of our starting pitchers."
Atlanta's Derek Lowe and Perez each gave up five hits and three runs in six innings. Perez walked four and struck out six, while Lowe walked three and struck out two.
Jeff Francoeur was 0 for 4 with an RBI in his return to Atlanta following Friday's trade to the Mets. He tipped his cap in response to the cheers from fans in right field in the first inning and received a loud ovation before lining into a double play in his first at-bat.
"It was a little bit of a distraction," Francoeur said of his return. "It was a different experience. It was fun."
Ryan Church, who came to the Braves in the trade, did not start. He came in the game in the eighth as a pinch-runner and defensive replacement for Garret Anderson, who had three hits, including a homer.
The Braves took a 2-0 lead in the second on back-to-back homers by Yunel Escobar and Anderson.
The Mets scored three runs in the fourth. Luis Castillo, David Wright and Gary Sheffield opened the inning with three straight singles to drive in the first run. Francoeur's groundout drove in Wright, and David Murphy added a run-scoring double.
Matt Diaz tied the game with a two-out double in the fourth. McCann, who singled, scored from first on a headfirst slide with his hand touching the plate just before the tag from Schneider.
"That's probably the first run I've ever scored on a double down the left-field line," McCann said. "I felt like if I slid feet first he'd have got me."
The Mets selected the contract of IF Angel Berroa from Triple-A Buffalo before the game. Berroa, 31, was signed as a minor league free agent on Saturday after he was designated for assignment by the Yankees on June 24. ... Escobar and Anderson hit the Braves' first back-to-back homers this season. ... Norton snapped an 0-for-11 slump with his first hit since June 5.
Giants aim to break 4-game slide, play the Mets
New York Mets (28-15, first in the NL East) vs. San Francisco Giants (22-18, third in the NL West)
Nationals begin 3-game series against the Dodgers
Los Angeles Dodgers (27-13, first in the NL West) vs. Washington Nationals (14-28, fifth in the NL East)
Astros start 3-game series with the Guardians
Cleveland Guardians (17-20, third in the AL Central) vs. Houston Astros (27-15, first in the AL West)
Yankees host the Orioles to begin 3-game series
Baltimore Orioles (17-25, fifth in the AL East) vs. New York Yankees (29-12, first in the AL East)
Mariners begin 3-game series with the Athletics
Oakland Athletics (17-26, fifth in the AL West) vs. Seattle Mariners (17-25, fourth in the AL West)
Reds take on the Cubs in first of 4-game series
Chicago Cubs (16-24, third in the NL Central) vs. Cincinnati Reds (12-28, fifth in the NL Central)